We elaborate on the proposed general boundary formulation as an extension of standard quantum mechanics to arbitrary (or no) backgrounds.Temporal transition amplitudes are generalized to amplitudes for arbitrary space-time regions.State spaces are associated to general (not necessarily spacelike) hypersurfaces.We give a detailed foundational exposition of this approach, including its probability interpretation and a list of core axioms.We explain how standard quantum mechanics arises as a special case.We include a discussion of probability conservation and unitarity, showing how these concepts are generalized in the present framework.We formulate vacuum axioms and incorporate space-time symmetries into the framework.We show how the Schrödinger-Feynman approach is a suitable starting point for casting quantum field theories into the general boundary form.We discuss the role of operators.
The usual formulation of quantum theory is based on rather obscure axioms (employing complex Hilbert spaces, Hermitean operators, and the trace rule for calculating probabilities). In this paper it is shown that quantum theory can be derived from five very reasonable axioms. The first four of these are obviously consistent with both quantum theory and classical probability theory. Axiom 5 (which requires that there exists continuous reversible transformations between pure states) rules out classical probability theory. If Axiom 5 (or even just the word "continuous" from Axiom 5) is dropped then we obtain classical probability theory instead. This work provides some insight into the reasons quantum theory is the way it is. For example, it explains the need for complex numbers and where the trace formula comes from. We also gain insight into the relationship between quantum theory and classical probability theory.